Continuous to Vibrating via Irregular Pinion: How It Works, Parts, Formula, and Industrial Uses

The carrier is part of the mechanism

The circular pinion centre moves on an arc about the lower pulley axis. As the irregular gear turns, its changing outline rocks that carrier. The horizontal slotted arm is rigidly attached to the carrier. Its pin remains on the vertical rod guide and slides along the slot as the arm changes angle.

The belt connects the upper and lower pulleys; their centre distance stays equal to the carrier length. Their displayed 3:1 radius ratio is an assumption of this model. Belt rotation is calculated relative to the moving carrier.

The coloured gear boundaries are pitch outlines. Tooth flank geometry requires further design beyond these pitch outlines. A green contact point marks the rolling contact used in the calculation.

Compare an oscillating output

Choose carrier length, nominal gear spacing, rocking amplitude and irregular-gear speed. Rod stroke, peak speed and rocking frequency follow the selected geometry. The circular pinion radius is calculated to close the pitch cycle at three turns per irregular-gear revolution.

This model illustrates the irregular-gear and rocking-carrier arrangement. It is not a capacity, vibration-isolation or tooth-strength calculator, and the speed range is a calculation input rather than an operating-speed rating.

Constructing the pitch envelope

Let θ be the irregular-gear angle and φ=A sin θ the selected carrier angle. In drawing coordinates, O=(0,0), C=(L sin φ,−L cos φ), and the irregular-gear shaft is G=(−D,−L). Let J rotate a vector by 90°, primes denote differentiation with respect to θ, and v=C−G+JC′.

Select circular pinion radius r=mean(|v|)/4. The contact point is K=C−rv/|v|. Rotating K−G by −θ constructs the fixed pitch outline of the irregular gear. The pinion angle derivative is ψ′=1−|v|/r. This satisfies equal velocities at pitch contact and gives exactly −3 full pinion turns per irregular-gear revolution.

The lower pulley angle is γ=φ+(ψ−φ)/3 for the selected 3:1 open belt. With rod-guide offset H=0.6L, rod displacement is y=−H tan φ and stroke is 2H tan A. Over the stated amplitude range, peak rod speed is HAω, where ω=2πn/60; rocking frequency is n/60.

The computed outline describes the rolling pitch boundary used to obtain the selected carrier motion. Tooth flanks, module, backlash, bearing forces and belt tension remain outside the model.

Change amplitude without stretching the carrier

At 200 mm carrier length and ±6° rocking, rod A has a stroke of about 25.225 mm. At 20 gear rpm, it completes one cycle every three seconds and its peak speed is about 26.319 mm/s.

Doubling input speed doubles output speed and frequency without changing stroke or the pitch geometry. Changing the nominal gear spacing alters the pitch geometry and calculated pinion radius; it does not change the prescribed rocking amplitude.

Motion assumptions and limits

The source establishes the irregular gear, pinion, rocking arm, belt and rod arrangement. It gives no dimensioned pitch or tooth law. The model assumes sinusoidal carrier motion, the stated length ratios, a 3:1 pulley ratio and three pinion turns per gear turn.

The pinion centre moves with the carrier, so its rolling speed depends on both carrier motion and gear rotation. A manufactured gear requires conjugate tooth design in addition to a valid pitch outline.

Irregular-pinion questions

Does the pinion reverse?

No, within the stated range it continually rotates while its carrier rocks.

Why does the rod pin slide?

A fixed vertical guide intersects the rotating arm at a changing distance from the pivot. The slot permits that movement.

Why is pinion radius an output?

It is chosen from the rolling envelope so the pitch cycle closes at three turns per gear revolution.

Are the coloured outlines actual tooth tips?

No. They are the calculated pitch boundaries. This page does not provide a fabrication-ready gear.